What should be the height of the conical tent ?

Question :

In a marriage ceremony of her daughter Poonam, Ashok has to make arrangements for the accommodation of 150 persons. For this purpose, he plans to build a conical tent in such a way that each person has 4 sq. meters of the space on ground and 20 cubic meters of air to breath. What should be the height of the conical tent ?

Solution :

Let the height of the conical tent = h metre.

Radius of the base of the cone = r meter.

The tent has to accommodate 150 persons.

The space required by each person on the ground = 4 $m^2$

And the amount of air = 20 $m^3$

$\therefore$  Area of the base = 150 $\times$ 4 = 600 $m^2$

$\implies$   $\pi r^2$ = 600 $\implies$ r = 13.817 m

Volume of the air required for 150 persons = 150 $\times$ 20 = 3000 $m^3$

$\implies$  $1\over 3$ $\pi r^2 h$ = 3000 $m^3$

$\implies$ h = $3000 \times 7 \times 3\over 22 \times (13.817)^2$ = 15 m

Hence the height of the conical tent is 15 m.